The Local Structure of Graded Representations

Bocklandt, Raf; Symens, Stijn

doi:10.1080/00927870600938365

Cited by 5 publications

(7 citation statements)

References 13 publications

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“…To connect the two notions above, a result of Le Bruyn [, Proposition 6 and its proof] and a result of Bocklandt and Symens [, Lemma 4] says that any simple

g

‐torsionfree module

V

over

S

corresponds to (as simple quotient of) a fat point module

F

of period

e

in such a way that

dim V = d e

with

d = mult (F)

, and the stabilizer of

V

{PGL}_{d e} false(double-struckk false) \times {double-struckk}^{\times}

is conjugate to the subgroup generated by

(g_{ζ}, ζ)

with

g_{ζ} = diag false(\underset{d}{\underset{︸}{1, \dots, 1}}, \underset{d}{\underset{︸}{ζ, \dots, ζ}}, \dots, \underset{d}{\underset{︸}{ζ^{e - 1}, \dots, ζ^{e - 1}}} false)

and

ζ

is a primitive

e

‐th root of unity. …”

Section: On the Representation Theory Of Smentioning

confidence: 99%

Poisson geometry of PI three‐dimensional Sklyanin algebras

Walton

Wang

Yakimov

2018

Proc. London Math. Soc.

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We give the three‐dimensional Sklyanin algebras S that are module‐finite over their center Z, the structure of a Poisson Z‐order (in the sense of Brown–Gordon). We show that the induced Poisson bracket on Z is nonvanishing and is induced by an explicit potential. The double-struckZ3×double-struckk×‐orbits of symplectic cores of the Poisson structure are determined (where the group acts on S by algebra automorphisms). In turn, this is used to analyze the finite‐dimensional quotients of S by central annihilators: there are three distinct isomorphism classes of such quotients in the case (n,3)≠1 and two in the case (n,3)=1, where n is the order of the elliptic curve automorphism associated to S. The Azumaya locus of S is determined, extending results of Walton for the case (n,3)=1.

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“…To connect the two notions above, a result of Le Bruyn [, Proposition 6 and its proof] and a result of Bocklandt and Symens [, Lemma 4] says that any simple

g

‐torsionfree module

V

over

S

corresponds to (as simple quotient of) a fat point module

F

of period

e

in such a way that

dim V = d e

with

d = mult (F)

, and the stabilizer of

V

{PGL}_{d e} false(double-struckk false) \times {double-struckk}^{\times}

is conjugate to the subgroup generated by

(g_{ζ}, ζ)

with

g_{ζ} = diag false(\underset{d}{\underset{︸}{1, \dots, 1}}, \underset{d}{\underset{︸}{ζ, \dots, ζ}}, \dots, \underset{d}{\underset{︸}{ζ^{e - 1}, \dots, ζ^{e - 1}}} false)

and

ζ

is a primitive

e

‐th root of unity. …”

Section: On the Representation Theory Of Smentioning

confidence: 99%

Poisson geometry of PI three‐dimensional Sklyanin algebras

Walton

Wang

Yakimov

2018

Proc. London Math. Soc.

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“…In order to better describe the difference between these 2 kind of points in Azu n A, one has to use the fact that A is graded. We will use [3] to explain the difference.…”

Section: Trep N Amentioning

confidence: 99%

“…[3] sets this in a GIT -setting: this difference is given by the fact that for a degree n fat-point module F and a chosen representative M of the C * -orbit determined by F , the stabilizer in PGL n × C * of M is trivial, while for the simple modules for which xyz is in the kernel of the algebra map, the stabilizer is not trivial. According to [3], this stabilizer for a simple module is always a finite cyclic subgroup of PGL n × C * .…”

Section: Trep N Amentioning

confidence: 99%

“…One of the tools available in commutative algebraic geometry to resolve singularities is the use of blow-ups. Similar to [3], where the quantum plane C −1 [u, v] was blown-up in the unique singularity, and [6], where the 3-dimensional Sklyanin algebras where blown-up in their unique singularity, we will use the construction of a noncommutative blow-up in section 6 to define an algebra B = A ⊕ It ⊕ I 2 t 2 ⊕ . .…”

Section: Introductionmentioning

confidence: 99%

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Geometry of representations of quantum spaces

De Laet

2014

Preprint

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The quantum plane A = Cρ[x, y, z] with ρ a root of unity has singularities in its representation variety trep n A and its center C[u,v,w,g] uvw−g n . Using the technique of a noncommutative blow-up, we prove that this technique fails in contrast to the 3-dimensional Sklyanin algebras if we want to resolve the singularities in trep n A. However, we will see that the singularity of the center in the origin can be made better using this technique.

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“…The varieties PM ssimp d (Q) are introduced and analyzed in [3] as moduli spaces of representations of Q up to a notion of graded equivalence.…”

Section: Definition 24 We Define the Projectivized Moduli Spacesmentioning

confidence: 99%